diff --git a/examples/custom_op_demo.py b/examples/custom_op_demo.py
new file mode 100644
index 000000000000..d9c3c8289505
--- /dev/null
+++ b/examples/custom_op_demo.py
@@ -0,0 +1,95 @@
+import torch
+import torch.utils.cpp_extension
+import torch_mlir
+from torch_mlir import run_pipeline_with_repro_report
+from torch_mlir.ir import BoolAttr, Context, Module, InsertionPoint, Location
+from torch_mlir_e2e_test.annotations import export, annotate_args
+
+
+def identity(_5: torch.Tensor):
+    return _5
+
+
+goofy_lib = torch.library.Library("goofy", "DEF")
+goofy_lib.define("identity(Tensor t) -> Tensor")
+goofy_lib.impl("identity", identity)
+
+
+class CustomOpExampleModule(torch.nn.Module):
+    def __init__(self):
+        super().__init__()
+
+    @export
+    @annotate_args(
+        [
+            None,
+            ([-1, -1], torch.float32, True),
+        ]
+    )
+    def forward(self, a):
+        b = 2 * a
+        return torch.ops.goofy.identity(b)
+
+
+mod = CustomOpExampleModule()
+mod.eval()
+
+module = torch_mlir.compile(mod, torch.ones(3, 4), output_type="raw")
+
+pipeline = (
+    "symbol-dce,"
+    "torch-prepare-for-globalize-object-graph,"
+    "torch-globalize-object-graph,"
+    "symbol-dce,"
+    "inline{default-pipeline= max-iterations=4 },"
+    "torch-adjust-calling-conventions"
+)
+
+run_pipeline_with_repro_report(
+    module, pipeline=f"builtin.module({pipeline})", description=""
+)
+print(module)
+
+forward = module.operation.regions[0].blocks[0].operations[1]
+goofy_op = forward.operation.regions[0].blocks[0].operations[4]
+goofy_op.attributes["has_value_semantics"] = BoolAttr.get(True, context=module.context)
+
+print(module)
+
+abstract_interp_src = """\
+func.func @__torch_mlir_shape_fn.operator.goofy.identity(%arg0: !torch.list<int>) -> !torch.list<int> {
+  return %arg0 : !torch.list<int>
+}
+func.func @__torch_mlir_dtype_fn.operator.goofy.identity(%arg0: !torch.int, %arg1: !torch.int) -> !torch.int {
+  return %arg1 : !torch.int
+}
+"""
+
+with Location.unknown(module.context) as loc:
+    abstract_interp_module = Module.parse(abstract_interp_src)
+    with InsertionPoint.at_block_begin(module.body) as ip:
+        shape_fn = abstract_interp_module.body.operations[0]
+        dtype_fn = abstract_interp_module.body.operations[1]
+        InsertionPoint.insert(ip, shape_fn.detach_from_parent())
+        InsertionPoint.insert(ip, dtype_fn.detach_from_parent())
+
+print(module)
+
+run_pipeline_with_repro_report(
+    module,
+    pipeline="builtin.module(func.func(torch-reduce-op-variants,torch-maximize-value-semantics))",
+    description="",
+)
+
+print(module)
+
+run_pipeline_with_repro_report(
+    module,
+    pipeline="builtin.module(torch-lower-to-backend-contract{backend-legal-ops=torch.operator decompose=true max-iterations=10})",
+    description="",
+)
+
+shape_fn.detach_from_parent()
+dtype_fn.detach_from_parent()
+
+print(module)
diff --git a/include/torch-mlir/Dialect/Torch/IR/TorchOps.td b/include/torch-mlir/Dialect/Torch/IR/TorchOps.td
index 889a29908de0..4d1176e4a469 100644
--- a/include/torch-mlir/Dialect/Torch/IR/TorchOps.td
+++ b/include/torch-mlir/Dialect/Torch/IR/TorchOps.td
@@ -829,7 +829,8 @@ def Torch_DerefineOp : Torch_Op<"derefine", [
 }
 
 def Torch_OperatorOp : Torch_Op<"operator", [
-    AllowsTypeRefinement
+    AllowsTypeRefinement,
+    HasValueSemantics
   ]> {
   let summary = "Opaque torch operator";
   let description = [{
diff --git a/lib/Dialect/Torch/Transforms/AdjustCallingConventions.cpp b/lib/Dialect/Torch/Transforms/AdjustCallingConventions.cpp
index 7cc699c04762..d4ac36a1c4e8 100644
--- a/lib/Dialect/Torch/Transforms/AdjustCallingConventions.cpp
+++ b/lib/Dialect/Torch/Transforms/AdjustCallingConventions.cpp
@@ -191,7 +191,10 @@ static bool isValidNonContainerResultType(Type resultType) {
          resultType.isa<Torch::FloatType>() ||
          resultType.isa<Torch::IntType>() ||
          resultType.isa<Torch::BoolType>() ||
-         resultType.isa<Torch::NoneType>();
+         resultType.isa<Torch::NoneType>() ||
+         (resultType.isa<Torch::ListType>() && cast<Torch::ListType>(resultType)
+                                                   .getContainedType()
+                                                   .isa<Torch::IntType>());
 }
 
 static LogicalResult validateReturns(func::FuncOp func) {
diff --git a/lib/Dialect/Torch/Transforms/ReifyAbstractInterpCalculationsUtils.cpp b/lib/Dialect/Torch/Transforms/ReifyAbstractInterpCalculationsUtils.cpp
index 04bbe0220e71..87085f9b8ffa 100644
--- a/lib/Dialect/Torch/Transforms/ReifyAbstractInterpCalculationsUtils.cpp
+++ b/lib/Dialect/Torch/Transforms/ReifyAbstractInterpCalculationsUtils.cpp
@@ -66,7 +66,16 @@ LogicalResult Torch::wrapWithCalculateOpIfLibraryFunctionAvailable(
         libFuncArgsBuilder) {
   Location loc = op->getLoc();
   MLIRContext *context = op->getContext();
-  auto name = op->getName().stripDialect();
+
+  std::string name_;
+  if (isa<OperatorOp>(op)) {
+    auto opOp = cast<OperatorOp>(op);
+    auto opName = opOp->getAttr("name").cast<StringAttr>().getValue();
+    name_ = "operator." + opName.str();
+  } else {
+    name_ = op->getName().stripDialect();
+  }
+  StringRef name = name_;
   // For value-semantic variant ops, i.e. valsem-ops (ops that are
   // mechanically consistent with existing torch conventions of in-place vs.
   //  out-of-place (value-semantic) variants), remove the prefix when
@@ -76,9 +85,17 @@ LogicalResult Torch::wrapWithCalculateOpIfLibraryFunctionAvailable(
   std::string libFuncName =
       (getLibraryFunctionPrefix(libFuncKind) + Twine(name)).str();
   auto libFunc = library.lookupSymbol<func::FuncOp>(libFuncName);
-  if (!libFunc)
-    return success();
-  libFuncNamesUsed.push_back(libFuncName);
+  if (!libFunc) {
+    auto parentModule = op->getParentOfType<ModuleOp>();
+    if (parentModule)
+      libFunc =
+          op->getParentOfType<ModuleOp>().lookupSymbol<func::FuncOp>(libFuncName);
+    if (!libFunc)
+      return success();
+  } else {
+    libFuncNamesUsed.push_back(libFuncName);
+  }
+
   OpBuilder b(op);
   Operation *calculate =
       createCalculateOp(b, loc, op->getResultTypes(), libFuncKind);